q-Polygamma Function

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The q-digamma function psi_q(z), also denoted psi_q^((0))(z), is defined as

[画像: psi_q(z)=1/(Gamma_q(z))(partialGamma_q(z))/(partialz), ]

(1)

where Gamma_q(z) is the q-gamma function. It is also given by the sum

[画像: psi_q(z)=-ln(1-q)+lnqsum_(n=0)^infty(q^(n+z))/(1-q^(n+z)). ]

(2)

The q-polygamma function psi_q^n(z) (also denoted psi_q^((n))(z)) is defined by

[画像: psi_q^((n))(z)=(partial^npsi_q(z))/(partialz^n). ]

(3)

It is implemented in the Wolfram Language as QPolyGamma [n, z, q], with the q-digamma function implemented as the special case QPolyGamma [z, q].

Certain classes of sums can be expressed in closed form using the q-polygamma function, including

[画像:sum_(k=1)^(infty)1/(1-a^k)] = [画像:(psi_(1/a)(1)+ln(a-1)+ln(1/a))/(lna)]

(4)

[画像:sum_(k=0)^(infty)1/(coshk+1)] = 2[1-psi_e^((1))(-ipi)].

(5)

The q-polygamma functions are related to the Lambert series

L(beta) = [画像:sum_(n=1)^(infty)(beta^n)/(1-beta^n)]

(6)

= [画像:sum_(n=1)^(infty)1/(beta^(-n)-1)]

(7)

= [画像:(psi_q(1)+ln(1-q))/(lnq)]

(8)

(Borwein and Borwein 1987, pp. 91 and 95).

An identity connecting q-polygamma to elliptic functions is given by

[画像: pi-i[psi_(phi^2)^((0))(1/2-(ipi)/(4lnphi))-psi_(phi^2)^((0))(1/2+(ipi)/(4lnphi))] =-(lnphi)theta_2^2(phi^(-2)), ]

(9)

where phi is the golden ratio and theta_n(q) is an Jacobi theta function.

See also

False Logarithmic Series, Polygamma Function

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

References

Borwein, J. M. and Borwein, P. B. "Evaluation of Sums of Reciprocals of Fibonacci Sequences." §3.7 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 91-101, 1987.

Referenced on Wolfram|Alpha

q-Polygamma Function

Cite this as:

Weisstein, Eric W. "q-Polygamma Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/q-PolygammaFunction.html

Subject classifications